An approach for the local exploration of discrete many objective optimization problems

International audienceMulti-objective optimization problems with more than three objectives, which are also termed as many objective optimization problems , play an important role in the decision making process. For such problems, it is computationally expensive or even intractable to approximate the entire set of optimal solutions. An alternative is to compute a subset of optimal solutions based on the preferences of the decision maker. Commonly, interactive methods from the literature consider the user preferences at every iteration by means of weight vectors or reference points. Besides the fact that mathematical programming techniques only produce one solution at each iteration, they generally require first or second derivative information, that limits its applicability to certain problems. The approach proposed in this paper allows to steer the search into any direction in the objective space for optimization problems of discrete nature. This provides a more intuitive way to set the preferences, which represents a useful tool to explore the regions of interest of the decision maker. Numerical results on multi-objective multi-dimensional knapsack problem instances show the interest of the proposed approach

A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning

We present a tutorial on Bayesian optimization, a method of finding the maximum of expensive cost functions. Bayesian optimization employs the Bayesian technique of setting a prior over the objective function and combining it with evidence to get a posterior function. This permits a utility-based selection of the next observation to make on the objective function, which must take into account both exploration (sampling from areas of high uncertainty) and exploitation (sampling areas likely to offer improvement over the current best observation). We also present two detailed extensions of Bayesian optimization, with experiments---active user modelling with preferences, and hierarchical reinforcement learning---and a discussion of the pros and cons of Bayesian optimization based on our experiences

Bayesian Optimization for Adaptive MCMC

This paper proposes a new randomized strategy for adaptive MCMC using Bayesian optimization. This approach applies to non-differentiable objective functions and trades off exploration and exploitation to reduce the number of potentially costly objective function evaluations. We demonstrate the strategy in the complex setting of sampling from constrained, discrete and densely connected probabilistic graphical models where, for each variation of the problem, one needs to adjust the parameters of the proposal mechanism automatically to ensure efficient mixing of the Markov chains.Comment: This paper contains 12 pages and 6 figures. A similar version of this paper has been submitted to AISTATS 2012 and is currently under revie

Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

Firefly Algorithm: Recent Advances and Applications

Nature-inspired metaheuristic algorithms, especially those based on swarm intelligence, have attracted much attention in the last ten years. Firefly algorithm appeared in about five years ago, its literature has expanded dramatically with diverse applications. In this paper, we will briefly review the fundamentals of firefly algorithm together with a selection of recent publications. Then, we discuss the optimality associated with balancing exploration and exploitation, which is essential for all metaheuristic algorithms. By comparing with intermittent search strategy, we conclude that metaheuristics such as firefly algorithm are better than the optimal intermittent search strategy. We also analyse algorithms and their implications for higher-dimensional optimization problems.Comment: 15 page